6. B.

As shown in Figure 2, connect $C F$.
By the perpendicular diameter theorem, we have
$$\text{A C = A D ,}$$

Thus, $\angle B=\angle A F D$.
Since $O C=O B$, we get
$$\angle B=\angle O C B \text {. }$$

Therefore, $\angle A F D=\angle O C B$.
Hence, $C, H, G, F$ are concyclic.
Thus, $\angle C G H=\angle A F C=\angle B$.
Consequently, $G H \parallel O B$.
In $\triangle O C B$, since $C G=G B$, we have $C H=H O$.
By the midline theorem of a triangle, we get
$$G H=\frac{1}{2} O B=\frac{1}{4} A B \Rightarrow \frac{G H}{A B}=\frac{1}{4} .$$